Make Math Moments Academy › Forums › Full Workshop Reflections › Module 3: Teaching Through Problem Solving to Build Grit and Perseverance › Lesson 34: Tool #2 – Multiple Representations › Lesson 34: Discussion Prompt

What was your big take away from this lesson?
How might you use the tool discussed?
What is something you are still wondering?

My big takeaway from this lesson is to take time to discuss multiple representations of a solution and for students to understand they all representations are appropriate in their own way.
I am wondering how I can apply multiple representations to other (possibly non algebraic) problems.

@steve For us, this is a huge part of teaching now. How can I bring out multiple strategies or models. You’ll definitely want to spend time in this arena as you develop your lessons. Anticipating multiple strategies to solve a problem is a great way to start.


The whole idea of moving from concrete to representational to abstract is another version of multiple representations. For operations with fractions, I know some students can visualize well with double numberlines while not following arrays very well. Others are the opposite. This is why we try to show models with both.

I always love relating the student representations to other student representations during the consolidation part of my lesson. I am always amazed by the different representations students come up with, most of the time there are representations I didn’t even think of. Thinking back to my high school and university days as a student I remember using procedures taught to me to expand and simplify expressions. I didn’t understand the purpose of it but enjoyed the challenge of it. This video was a fantastic way to see how all of the representations were shown algebraically and could be simplified to the expression n^2+2n+1 which is hard to relate to the original pattern. Also, I saw the growth of the pattern using the second example but really like seeing the other 2 examples in a visual way.
PS Love the tshirt. Was your daughter able to solve the pattern for Thursday?

Amazing points here.
Yes, she eventually got Thursday after some purposeful questioning!!


I love doing those types of patterning activities. I included one as a part of a weekly routine last year. I was so shocked when one of my 6th graders wrote a quadratic equation for one of the patterns with no prompting. It helped me to understand how making any critical thinking process a routine that we did regularly was powerful and provided many learning opportunities.
I also like how using multiple representations helps to make the problem accessible to more students. It feels like the most basic way to differentiate without them knowing what you are doing to help access the learning and sense making.

The big idea is to allow students to see that even when ideas look different, they can typically be manipulated to be the same as other thinkings. I have always enjoyed studying geometric patterns with my students because they can make their generalizations and predictions based on evidence that they find or create. The Heart problem is fantastic because it is not a problem that will become clear at first glance. Students can use manipulatives and draw diagrams to represent the pattern’s growth. However, if students find a working generalization of the design, I can always follow up with, “Can you find the pattern another way?” Great lesson.
I honestly do not currently have a wonder.

Great reading your reflections and hearing so many advocates for making connections between multiple representations. We hear our experience as a student echoed in some of your comments including the idea of knowing steps and procedures but never realizing there were other ways to see it. So important to reach all students!

My big takeaway is that I could not get this pattern without seeing a represnetation. The old linear slope/intercept method was not helping me. Then the representation I used led me to a brick wall. It reminds me of this video I showed my students at the beginning of the year about seeing scenarios from different perspectives and the importance of creating this habit/culture inside and outside the math classroom: https://www.youtube.com/watch?v=_SsccRkLLzU
I wonder the best way to create this culture. I like the ideas of gallery walks (wondering how I can do that remotely…maybe Google slides where everyone has a slide or a Jamboard…). I have not done a great job of creating this classroom culture yet, but these PDs are helping me do it slowly.
I also wonder how you introduce new ways to represent it students do not come up with them without either being the “gatekeeper” of knowledge or just living without a key representation. Sometimes I drop an idea to one group and then let them present it as their own but not sure that is always best.

Great video and that is such a great summary of what visuals and multiple representations can do in math class!
As for nudging students along if an idea doesn’t emerge: it is all about purposeful questioning. Rather than dropping an idea and potentially robbing the thinking from students, asking questions that might focus or funnel (depending on the situation) can be helpful. It takes a lot of practice, but questioning can really improve the experience!


Multiple representations means leaving space for a different viewpoint…asking ourselves, “How can I turn what I think I see on its head?” and “What questions can I ask others to better understand their view?” And further, I have to remember the strategy of asking a question and then WALKING AWAY!

Different representations for me is really important, one of the first goals in my classes is to make understand that in math and also in live there are always many ways to do things or to think about …
So, I always show the different strategies that the others have done. But I’ve always trouble to maintain the whole students attention when we are showing or talking about the strategies of their partners. They are happy with Their own strategy, that they have shown in the blackboard and they are not interested with the others. I hope that, maybe the clue Is that I have to prepare one more view to see the problem and ask them to work further with that. And then, if they wont to get right with the view I propose, they will know that if they are attentive on the others views they will be successful in my proposition.

That’s a great strategy Laura. I often use that. After we highlight a strategy from the class asking the rest of the class to go back and try that strategy on the current problem or the next problem can help with attention. Also, reminders about etiquette and classroom norms help as well.


I find multiple representations exciting and they give a glimpse into the beauty of math and different perspectives that are each valid. So far in my own math learning journey, I sometimes have difficulty thinking of more than 2 representations. Does this get better over time?

I had two big takeaways from this lesson. First, I couldn’t believe you made your mom go through a productive struggle. I mean who withholds information from their mom in the first place?? Second, was that I never would have gotten the third representation if I had not seen it broken down visually. I recognized the first one immediately, understood the second visually and algebraically, and would have been in the dark on the third without the representation. I often wrap up lessons by having students present multiple pathways to their solutions, but I would like to develop multiple representations more in my build up toward the final goal. Also, it was really easy to follow this on the Hero Model. I recognized the Withholding, Predict, Productive Struggle, and Consolidate as they happened.

Sounds like a lot of big take aways for you on this lesson! awesome!
And always remember: EVERYONE deserves a good productive struggle… even mom! 
I came up with a fourth representation
Wednesday can be thought of as a 3×3+5 extra hearts, Thursday would have a 4×4+7 extra, Friday would be a 5×5+9 giving a nxn + (next consecutive odd # or 2n1)


Honestly, my big take away is that I need another teacher in the room with me for this one! I am not good at this at all! I can come up with some of this on my own, but it would be tough for me to lead the discussion. Continued practice is needed! This is pretty intriguing.

Coteaching is so helpful if you ever have the opportunity. It truly is impossible to see/hear and make sense of what is happening in the room all while trying to facilitate at the same time. Over time we get better at these things, but having another set of eyes and ears is so helpful.


I encourage students to look for different ways of approaching the problem, but sometimes students say that the approach that is less common is too confusing and the student who presented that idea is left feeling that his solution is somehow inferior. I loved you idea that learning to see multiple representations will help us see different perspectives when communicating with others. It takes this out of the math realm and into a social skill.

Yes! And it is important for us to try to select and sequence student approaches that we can connect to other representations and namely the representations we are trying to highlight that day. This might require us being the facilitator and explicitly connecting those ideas – otherwise everyone in the room leaves feeling a bit less confident including the sharer.


I think different representations is important. I always tell my students that they can get the answer any way they want as long as they show me their work. We have many discussions throughout the year about why some representations may be better than others because sometimes when we try to draw all the pieces it is easier to make mistakes. I have had 3rd graders who will represent 57 + 23 by drawing 57 circles and 23 circles. We talk about how we might make a counting error when we draw that many circles and then talk about other ways we might represent this (draw 5 tens and 7 ones and 2 ten and 3 ones).

Love it. The need for changing our model / tool should be a natural fit. “My numbers are too big, I’ll move to a more symbolic representation”.


The highlight in this lesson is the beauty of multiple representations. To a large extent the fact that the point of view might differ, however, the information conveyed remain the same. This provides learners multiple opportunities to take in the information
Armed with that realization, I suppose the challenge is to find ways to include multiple representations in my lessons.
I wonder if there is any concept that multiple representations does not apply.

Great question re: whether there is ever a time where multiple representations doesn’t apply. I’m going to argue there is always multiple representations possible… however sometimes they may look more similar than others. I find that when we allow students to solve problems without preteaching a specific method, they come up with many new approaches that I would never have thought of…


My big takeaway was that there are many ways to look at a problem. I already do this a lot when we do number talks and dot talks at the beginning of class. My students know that I am going to want a different way to do something. I can see that it would be helpful to do more of this in the other assignments during class as well. I looked at the dot pattern and saw one rectangle that was n1 by n+1 and another rectangle that was 2 by n. It is great to find different ways to look at things.

Fantastic to hear! The part that takes a long time to build up the pedagogical moves for us trying to unpack those visual patterns and connect them to the symbolic representations. Fun times indeed!


I liked how working to find the pattern encouraged me as a learner to consider how to represent my thinking in ways that were more efficient to solve for much higher days. I personally enjoyed thinking about the task as a puzzle, and can imagine how students will enjoy this on the same level.
As a teacher, I was thinking about the types of “teacher moves” I would want to make in order to support students as they are searching to find ways to make sense of the math used to solve these types of problems. I like that students are provided the opportunity to invest in solving a tricky problem with just being given rules and formulas.
I can see how this sort of activity sparks interest in the mathematics so that we can spend time consolidating learning with various types of problems over time.

Multiple representations are the bread and butter of consolidation as I see it and mathematics in general. Teaching multiple representations allows students to find the strategy that makes sense for them. I like it because each representation is a window into that student’s thinking and making that thinking visible.

Bread and butter is right! They are the glue that fuels sense making and holds all of the mathematical understanding together.


What was your big take away from this lesson?
importance of asking questions, and more questions
never shut down an idea
keep prompting them
How might you use the tool discussed?
demonstrate all of the different ways of thinking that students present to show there are multiple ways to solve and tackle problems
What is something you are still wondering?
– how far do you push your students with showing multiple ways, stronger kids often push back and do not want to show their work just their answer.

Great take aways and wonders!
When it comes to making connections across representations note that while students can all enter using their own student generated approaches, we can’t highlight them all. We want to specifically select and sequence them to help emerge the big ideas you are after. This means that you can be more focused rather than turning the consolidation into a big “show and tell” session that is long and disconnected.
Also, we can’t force representations on them. If they aren’t developmentally there yet, we don’t want to push towards a model and or strategy that is too abstract or disconnected.


Multiple representations allow students to enrich their understanding and make stronger connections to relationships or how things are connected and that there may not be just one specific way to go about “answer” but many ways to do so.
My biggest worry is not being able to see more than 2 ways to view a problem/solution and I may not anticipate all the ways students may come up with them. Then again, learning along with my students is a good thing. Different lenses can often open your mind/eyes to different ways of doing something.

I was so bought in during that whole video! I was like cheering by myself when I figured out the same representation as you! I can see students loving the multiple representations as a way to really understand math. The most important thing for me is to create multiple representations in advance of the lesson so I am prepared and more familiar. Confession… sometimes I’m just planning on the fly and don’t always look at all angles ahead of time!

One of the most important things we can do is anticipating student solutions / approaches. This makes our consolidation much more intentional and we have less (not none) situations where you will have to think on your feet and/or adlib.

Anticipating student answers is still a challenge for me but I’m getting better.


Having students find multiple representations allows them to move more fluidly between them. If we honour more representations we honour more ways of thinking building a more responsive culture in our classroom.

When I first saw “Multiple Representations” as a tool, I thought it meant using patterns, graphs, tables, and/or equations being the different representations, instead of multiple ways of “seeing” the same thing using the same method of representation. Based on this, my biggest take away is how important it is for students to first work independently on this type of task before having them share visually and verbally how they saw it, with a partner and then a team, prior to a wholeclass discussion.
I plan on providing more time in class for students to do the exploration and sharing of patterns before moving into writing expressions algebraically and usually only in one way of seeing it. By doing this and showing that students are looking at the same pattern but just “seeing” the pattern differently when they explain it, they actually are all seeing the same pattern algebraically as well. I think this will also deepen students’ understanding of the distributive property and combining like terms.

Being someone who finds one way to do something and then struggles to find a different way to look at it makes anticipating student methods tough, but it forces me to stay open to all of their ideas and I find that when I question them, asking HOW their representation works, we all come away with better understanding. (Some days, I walk out thinking I learned more than my students did!!)

Love this! You’re right, it’s possibly you can’t come up with a new way…..but when your students do you get to file that solution away for the next time you teach that topic.


The beauty of Algebra is that there is more than one way to approach a problem, and having students argue for their method, before they realize that their equations are equivalent is so fun!

Multiple representation is important for understanding. I, myself, have struggled with the visual representation…I always tend to want to figure this type of problem with algebra. Definitely need more practice solving these types of problems with visuals!

I love how you encouraged learners to draw it out whether they were correct or incorrect.

My biggest take away from this lesson is that I am glad I have been having students show multiple representations whenever possible. It is one of the best ways to assess the wide variety of mathematical depth of understanding that exists in a single classroom.
I like to present an openended problem to students and give them time to talk about the problem with a partner or two to make sure they understand what is being asked. Then I give them time to represent their own solutions. Then they are encouraged to share their own solutions with classmates who solved it a different way. Eventually, we share the different solutions to the whole group, so kids get to see several possible ways to solve the same problem. This is a great way for kids to add to their own repertoire.
One thing I’m still wondering is whether I am giving them enough opportunity to do this.

Allow students to find multiple representations when solving problems is my big take away from this lesson. This is what I struggled with the most during the last school year (my first year teaching). I was so focused on following the textbook that I neglected to permit my students to show and share their thinking. As a result, they were often disengaged and/or forgot what strategies they were taught. If I had taken the time to slowdown, I would have seen the value of exploring their thought process. This goes back to y’alls riding a bike analogy.

My biggest take away from this is that multiple representations isn’t just about math! It is about life (and math is way to represent life). Being able to see things from multiple perspectives helps us build empathy and become better people, as well as better mathematicians!

I think this will be my hill to climb in helping my students. When I began teaching grade 3 only a few years ago, I was stuck in the algorithms I had learned years ago. Though I used manipulatives, I had a long way to go in seeing the flexibility in math. In the last 23 years I have been pursuing this more flexible understanding of math and have grown. Many students still want me to tell them how before they solve and I know I have still fallen into that time and again.
The examples here as well as the contributions of my students in the last school year are helping me embrace flexibility. I still wonder, though, since I am teaching in the early years, where does teaching different forms of representation come into this process? If I just leave it open, they tend to watch the students who know the “sought after algorithm” to “solve it easier”. I am hoping by starting this process with patterning using objects and shapes, I can develop a mindset of flexible thinking before I get into numbers.

Sometimes I feel due to my own “traditional” math education, where I would not have considered myself a “Math Person”, it is hard for me to figure out multiple ways…in your example, I figured it out the second way but did not look at the arrays to do it – I looked at rows and columns and what happened to the rows and then what happened to the columns and how I could represent that with N…I guess it was arrays in a way, but that was not my thinking. I was thinking rows and columns. My takeaway is if a student gets an answer it is important to look at their thinking since they might have looked at something very different than how I looked at it.

Multiple representations are so important because we all see things in different ways. I solved your problem with your second equation. When you gave the first one, I started thinking that I did it wrong. Then you showed the second way. I think I felt this way because of a mindset of one right way to do the math. I need to make sure that I am not creating that for my students. I want them to always think that there are other ways to solve a problem and for them to want to try and find them. I also know that our state test likes to test items using multiple representations, so it is important to provide that for our students.

Presenting students with multiple representations allows them to find at least one way that they “understand” what is happening. Requiring them to use more than one strategy further solidifies their understanding. I had a student who would take FOREVER to solve her long division problems because that’s where she was with her knowledge. Other students would have her show them how she solved it with her visual/representation and most didn’t get it. One day it finally clicked and the faster method made sense and she was able to represent it and the class clapped for her. She said that it was so much faster and she was glad that she had another way to solve that type of problem even if it did take a long time. These various representations are sometimes what allows a struggling student to find success.

What was your big take away from this lesson?
A: By giving hints you stifle their ideas.How might you use the tool discussed?
A: Don’t “Answer” their questions, instead ask questions to further their thinking.What is something you are still wondering?
A: In a quadmester, for example, can you be so liberal with your time?
Also, you have “good” and “notsogood” at math kids, what’s a good equity line to not make one group bored and not too fast for the other group? 
I always use patterns and puzzles with my students, to make them think, to get them talking about their thinking and of course problem solving. I like to build the level of difficulty and push students to think, be creative. I love this puzzle and use similar ones in my classroom. I love giving problems of the week that really challenge students but usually have different levels so I can reach all students.

So often my students, and yes even I as the teacher find “one way that works” and then stop. I have been the one that “shows other ways to do things” but I haven’t encouraged stretching and finding lots of ways to represent what is going on in a situation.

This is certainly hard work, but at least you’re aware of it and you can think of how you might be more intentional about highlighting multiple representations / approaches. Also, consider whether or not you’re preteaching before students problem solved as that will lead to more mimicking behaviour than true problem solving.


My biggest take away was that multiple representations don’t mean just show me the graph, equation or table for an solution….showing how concrete manipulatives can be used to illustrate the same solutions will help students believe that there are truly multiple ways to solve problems and that each has value.
 This reply was modified 2 months, 4 weeks ago by Mary Jackson.

I love the idea of using this to generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties. It also covers combining like terms and then it could move into algebraic equations. I always tell my students to draw pictures and this is using something they’ve seen before, area. This combines the exploration of so many standards into one lesson.
I am still wondering though, how long should I allow for this type of discovery to happen? Sixth grade math standards in Texas are so abundant that we struggle getting through them in a year. I have had a strong belief for quite some time that sixth grade is less about mastery and more of an introduction. It is their first year thinking abstractly in math and they see so much that will be mastered in later years. Is this something I could allow them to play with once and then reference back? And what about the students that can not make the connections. Would you say this is good for ALL students or more of the advanced students???

Wow! I was blown away at the end when all of the algebraic expressions were the same. I show arrays of dots at the beginning of the year and have kids describe how they figured out how many. Some count each one, some by twos, some see groups, etc. but we talk about how the answer is the same, but we get there different ways. (Grade 3) What works for one student visually doesn’t always work for another.

I love this!!! Our students are always thinking about problems in different ways and I love how this approach to problem solving lets them know it’s okay to think creatively. I love how showing your parents a visual helped gently guide them in the right direction, but it was still open ended in that they still had to figure out the pattern independently. I can’t wait to have my students explore these patterns in the beginning of the year and then come back to them after a lesson on multiplying binomials to show how each expression can be simplified to the same expression.

My big take away was the importance of multiple representations. I solved it differently than the video in the first example but the dialogue that accompanies the task left space for that. Really liked the challenge of finding another way to represent the solution. I am wondering if this resource is a part of a larger compendium.

Multiple representations! I love using this in our fractions units but I really like how the video showed ways of encouraging individuals when they just want the answer. I think that will be my biggest takeaway for this section.

So glad to hear it! Keep on promoting those multiple representations and keep student thinking going!


My big takeaway from this is that using multiple representations is a great way to reach students for whom a more traditional approach isn’t working. I have to be honest and say that I struggled with this, as right away I just found a pattern in the numbers without the visual and wasn’t sure how to make sense of the visual. I think the students might be better at that part than me…my brain just went straight to Algebra! I would ask my students to come up with different ways to represent the pattern, and know I would get a variety of responses. Last year when we were working through a unit on finding the surface area of polygons, I was amazed at the number of ways my students found to decompose a single polygon into shapes to find the surface area.

Wow! This was beautifully executed, Kyle. Thank you.
We will all find our most comfortable way of representing a problem to solve it. I love that about my students because there are a few who will often find a more abstract or unexpected way to solve a problem that was not even in my line of sight. Sometimes I have to work harder than them to stay ahead of what they’ll throw into the ring. There is never a dull day, and the students love to learn.